By H. Seifert, W. Threlfall

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Computation of H*(I>f) Case 1 ) : Topologie. o'sa k 2 Als o, We have 0 si a2 In the process described there, each sequence of four diagram, into We label the edges of the i-th handle-producing sequence 6 of Seifert and Threlfall 's Lehrbuch der consecutive edges, oriented and identified as in the following M is obtained by subdividing a'2 Each handle-producing sequences and We will assume that the re ader is familiar with the fact -2 a ... " = o'sa . = eSb. = 0 . 1 1 morphic to a direct sum of Case 2 ): k 1.

Ln' q, q Since of First we prove the following theorem. are ~he spaces of sub complexes IKI Then %- 70:0 + 6X(M) > o . te~ K is a regular complex, and if the connected 2 - simplices, tao - 20:1 Thus Homology and Connectedness zation of the number of connected components of a space in i s a finite 2-dimensional simplicial complex, 6X(M) = ,the binomial coefficient . out boundary) every I-simplex is the face of exactlY so h(JP2) they will provide some information about the topological properties edge s , etc .

Oa together with the boundary homomorphisms Thus homology with This is the main result of this section. K. 4. DEFINITION. {Cq(KjG)} Z . rith coefficients in G It turns out, however, =0 . f icients in an arbitrary abelian group Cq(KjR) ~ induces boundary homomorphisms mapping The proof of these facts is are all isomorphic. Homology with general coefficients is useful in some cases. If as defined previously. a Hq(KjG) are isomoD'hic, and that the homology Ca(K jG) easy and will be omitted. With this boundary operator the form a chain complex denoted by = Ca (K) As in the previous section, one can show that all of the because of is the chain which takes K[a:T]T, where of elements of chain modules q More precisely, if the value 1 on of R) we set (/1: C (KjR) An elementary chain complex is a chain complex Z of one of the following three types: 59· i) (Free) All chain groupS are zero except ~or an in~inite ••• cyclic group in a single dimension.